21 lines
1.2 KiB
TeX
21 lines
1.2 KiB
TeX
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\chapter{Crystallographic orientations}
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\section{Bunge Euler angles}
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\label{bunges}
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Euler angles (\beaphiI, \beaPhi , \beaphiII) -- in Bunge convention -- rotate the sample coordinate system ($X$,~$Y$,~$Z$ or RD,~TD,~ND) into the crystal coordinate system ($x_c$, $y_c$, $z_c$).
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Three successive rotations are carried out in the following way \citep[pg.\,4]{Bunge1982}:
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\begin{enumerate}
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\item Rotate by angle $\pI$ around Z, to bring X into the $x_c$--$y_c$-plane. The new intermediate axes are X', Y' and Z (Z is unchanged).
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\item Now rotate $\beaPhi$ degrees around X', to make Z parallel with $z_c$. The intermediate axes are X', Y'', Z'.
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\item A rotation by angle $\pII$ around Z' makes the rotated axes then identical to the crystal axes.
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\end{enumerate}
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The rotation matrix can be calculated as
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\[% Gottstein pg 55
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\bsym{g}=\left(\begin{array}{ccc}
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\cos{\pI}\cos{\pII}-\sin{\pI}\sin{\pII}\cos{\beaPhi} & \sin{\pI}\cos{\pII}+\cos{\pI}\sin{\pII}\cos{\beaPhi}& \sin{\pII}\sin{\beaPhi}\\
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-\cos{\pI}\sin{\pII}-\sin{\pI}\cos{\pII}\cos{\beaPhi} & -\sin{\pI}\cos{\pII}+\cos{\pI}\cos{\pII}\cos{\beaPhi}& \cos{\pII}\sin{\beaPhi}\\
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\sin{\pI}\sin{\beaPhi} & -\cos{\pI}\sin{\beaPhi}& \cos{\beaPhi}
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\end{array}\right)
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\]
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